Question: Solve for $x$, $ \dfrac{1}{2x - 2} = \dfrac{2}{x - 1} + \dfrac{2x + 10}{x - 1} $
Explanation: First we need to find a common denominator for all the expressions. This means finding the least common multiple of $2x - 2$ $x - 1$ and $x - 1$ The common denominator is $2x - 2$ The denominator of the first term is already $2x - 2$ , so we don't need to change it. To get $2x - 2$ in the denominator of the second term, multiply it by $\frac{2}{2}$ $ \dfrac{2}{x - 1} \times \dfrac{2}{2} = \dfrac{4}{2x - 2} $ To get $2x - 2$ in the denominator of the third term, multiply it by $\frac{2}{2}$ $ \dfrac{2x + 10}{x - 1} \times \dfrac{2}{2} = \dfrac{4x + 20}{2x - 2} $ This give us: $ \dfrac{1}{2x - 2} = \dfrac{4}{2x - 2} + \dfrac{4x + 20}{2x - 2} $ If we multiply both sides of the equation by $2x - 2$ , we get: $ 1 = 4 + 4x + 20$ $ 1 = 4x + 24$ $ -23 = 4x $ $ x = -\dfrac{23}{4}$